Prime Number Formed by Concatenating Consecutive Integers down to 1

Theorem

Let $N$ be an integer whose decimal representation consists of the concatenation of all the integers from a given $n$ in descending order down to $1$.

Let the $N$ that is so formed be prime.

The only $n$ less than $100$ for which this is true is $82$.

That is:

$82 \, 818 \, 079 \, 787 \, 776 \ldots 121 \, 110 \, 987 \, 654 \, 321$

is the only prime number formed this way starting at $100$ or less.

Proof

Can be determined by checking all numbers formed in such a way for primality.

Historical Note

Richard K. Guy reports in his Unsolved Problems in Number Theory, 3rd ed. of $2004$ that this result was determined by Charles Nicol and Michael Filaseta, but he provides no citation and it has not been corroborated.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $82,818,079,787,776,757,473,727,170 \, \ldots \, 10,987,654,321$

• 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.)